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We verify this conjecture in some special cases, focusing especially on the case where F is additionally required to be k-uniform and λ is small.. In light of 1.2, it is interesting to n

On a Conjecture of Frankl and F¨ uredi

Submitted: Nov 8, 2010; Accepted: Feb 27, 2011; Published: Mar 11, 2011

Mathematics Subject Classification: 05D05

Abstract Frankl and F¨uredi conjectured that if F ⊂ 2X is a non-trivial λ-intersecting family of size m, then the number of pairs {x, y} ∈ X2

that are contained in some F ∈ F is at least m2
[P Frankl and Z F¨uredi A Sharpening of Fisher’s Inequality Discrete Math., 90(1):103-107, 1991] We verify this conjecture in some special cases, focusing especially on the case where F is additionally required to be k-uniform and λ is small

1 Introduction

least one member of F ,

∂iF :=



i

 : E ⊂ F ∈ F



F1, F2 ∈ F The well-known Fisher’s Inequality states that if F is a λ-intersecting family

of size m, then |∂1

con-jectured a similar inequality for |∂2

F | Conjecture 1.1 easily implies Fisher’s Inequality since |∂12F|
≥ |∂2

F | ≥ m2
proves |∂1

F | ≥ m

m If there does not exist x ∈ X such that x ∈ F for all F ∈ F , then

|∂2

2



∗ Department of Mathematics, University of California San Diego, La Jolla, CA, 92093, USA E-mail: anchowdh@math.ucsd.edu

Frankl and F¨uredi [7] verified Conjecture 1.1 when λ = 1 While this paper appears to

be the first to consider Conjecture 1.1 since [7], several special cases of Conjecture 1.1 had already been proved before [7] was published For example, Ryser [13], Woodall [16], and Babai [1] showed Conjecture 1.1 is true when m = n Majindar [11] proved Conjecture 1.1 for regular λ-intersecting families

deg(x), is defined to be the number of sets in F that contain x We can delete x ∈ X with

F | = |X| = n We say F is r-regular if deg(x) = r for all x ∈ X We say that F is a sunflower if deg(x) ∈ {1, |F |} for all x ∈ X We say

F is trivial if there exists x ∈ X with deg(x) = |F |, and is non-trivial otherwise For

a subset S ⊂ X we define the co-degree of S, denoted codeg (S), to be the number of

F | = m

F | ≤ m k2
< m

2
if m > k(k − 1) + 1

We note that Conjecture 1.1 is equivalent to the seemingly stronger statement that if

F | ≥ m2
Fisher’s inequality and its variants are usually proved by linear independence arguments [2]; one difficulty in proving Conjecture 1.1 in this way is understanding how to interpret the non-triviality restriction in a linear algebra setting

The main results of this paper verify Conjecture 1.1 in the following cases We will use Theorem 1.2 to prove Theorem 1.3

X

F ∈F

|F | 2



x∈X

deg(x) 2



2



then |∂2

|∂2

(i) If λ = 2, then |∂2

F | ≥ m2
and equality holds if and only if F is a symmetric design (ii) If λ = 3 and k /∈ {8, 11}, then |∂2

F | ≥ m2
and equality holds if and only if F is a symmetric design

In light of (1.2), it is interesting to note that Stanton and Mullin [14] once conjectured

this conjecture been true, Theorem 1.2 would have implied that Conjecture 1.1 is true for uniform families as well as characterized the case of equality Unfortunately, Hall [10] proved that Stanton and Mullin’s conjecture is true only for λ ∈ {1, 2} and produced counterexamples for every λ ≥ 3

Since (1.1) and (1.2) are equivalent for uniform families, Hall’s proof of Stanton and Mullin’s conjecture for λ = 2 shows that (1.1) is true for uniform, non-trivial, 2-intersecting families Combined with Theorem 1.2, Hall’s result proves Theorem 1.3 (i)

If (1.1) were true for every non-trivial 2-intersecting family, then Theorem 1.2 would im-ply that Conjecture 1.1 is true for λ = 2 We exhibit one non-trivial 2-intersecting family that does not satisfy (1.1), but feel that this may be the only counterexample Ryser [13] showed that there is a unique non-uniform 2-intersecting family with size m = n:

ˆ

F := {{1, 2, 4}, {1, 4, 6, 7}, {1, 2, 5, 7}, {1, 2, 3, 6}, {2, 3, 4, 7}, {1, 3, 4, 5}, {2, 4, 5, 6}}

F ∈ ˆ F

|F |

non-trivial 2-intersecting family for which (1.1) does not hold

then (1.1) holds

an argument similar to that of de Bruijn and Erd˝os [4] This summarizes the proof of Conjecture 1.1 when λ = 1 as the proof of Theorem 1.2 is trivial in this case

Theorem 1.2 implies that a uniform counterexample to Conjecture 1.1 is also a terexample to Stanton and Mullin’s conjecture It is not difficult to see that Hall’s coun-terexamples to Stanton and Mullin’s conjecture do not give councoun-terexamples to Conjec-ture 1.1; for definitions see [10] Hence, we can view ConjecConjec-ture 1.1 as a weakening of Stanton and Mullin’s conjecture

Another weakening of Stanton and Mullin’s conjecture is Conjecture 1.5, which is due

to Hall [10] Together with Theorem 1.2, we see that Conjecture 1.5 would imply that Conjecture 1.1 is true if F is additionally required to be k-uniform and k is sufficiently

such that if

It is natural to wonder if the obvious analog of Conjecture 1.1 for higher shadows holds By considering λ-blowups of projective planes of order q when q is large enough,

we have infinitely many nontrivial λ-intersecting families F satisfying |∂iF | < mi
for

2 Proof of Theorem 1.2

We use linear programming duality to prove Theorem 1.2 We will use Theorem 1.2 to prove Theorem 1.3 in Section 3

F |

codeg ({x, y}) = i, and observe that the following identities hold

X

i≥1

F ∈F

|F | 2



i≥1

 i 2



2

m 2



{x, y} ⊂ F The second follows from counting pairs ({x, y}, {F1, F2}) where {x, y} ∈ X2
, {F1, F2} ⊂ F , and {x, y} ⊂ F1 ∩ F2 Consequently, (a1, , am) is a feasible solution to

F |:

Minimize

m

X

i=1

i≥1

F ∈F

|F | 2



X

i≥1

 i 2



zi =λ

2

m 2



zi ≥ 0, i ∈ {1, , m}

The dual of this linear program is:

2

m 2



F ∈F

|F | 2

2



x + iy ≤ 1, i ∈ {1, , m}

The feasible region of the dual linear program (2.4) has extreme points given by

− j+11

2

,

2

j + 1

!

If F satisfies (1.1), then setting j = λ − 1 in (2.5) yields

|∂2

2

m 2



− 1λ

2

!

F ∈F

|F | 2

 2 λ



2



as desired Finally, note that the equality in (1.1) follows from counting pairs (x, {F1, F2}) such that {F1, F2} ⊂ F and x ∈ F1∩ F2

F | = m2

if and only if F is a symmetric design Ryser [13], Woodall [16], and Babai [1] showed

F | = m2

with codeg ({x, y}) = i We will show that F is k-regular, which immediately implies that F is a symmetric design By (2.6), we see that equality holds in (1.2), (a1, , am)

is an optimal solution to the primal linear program (2.3), and (2.5) with j = λ − 1

is an optimal solution to the dual linear program (2.4) By complementary slackness,

i ∈ {1, m} The constraints in the primal linear program (2.3) imply aλ−1+ aλ = m2
and λ−12
aλ−1+ λ2
aλ = λ2
m

2
, so aλ−1 = 0 and aλ = m2

λ|∂1

L(x)| in terms of deg(x) that will allow us to prove

codeg ({x, y}) = i and observe that the following identities hold

X

i≥1

i≥1

 i 2



bx,i = (λ − 1)deg(x)

2



The second follows from counting pairs (y, {F1, F2}) where {x, y} ∈ X2
, {F1, F2} ⊂ F , and {x, y} ⊂ F1 ∩ F2 Consequently, (bx,1, , bx,m) is a feasible solution to the following

L(x)|:

Minimize

m

X

i=1

wi

i≥1

iwi = (k − 1) deg(x) X

i≥1

 i 2



2



wi ≥ 0, i ∈ {1, , m}

The dual of this linear program is:

2



y + (k − 1) deg(x)z

2



y + iz ≤ 1, i ∈ {1, , m}

Since (2.5) with j = λ − 1 is a feasible solution, using (2.7) yields

k − 1

1

deg(x) − 1 2

λ 2

−1!

k2

x∈F

so deg(x) = k for all x ∈ X Hence F is k-regular and is thus a symmetric design

3 Proof of Theorem 1.3

In light of (1.1) and (1.2), we are interested in upper bounds on the sizes of non-trivial λ-intersecting families F that depend only on the sizes of the sets in F One of the first results of this kind is Deza’s theorem [6], which bounds the size of λ-intersecting families

in (3.8) is bigger than the upper bound on m in (1.2) by a factor of roughly λ

Since non-triviality is a stronger restriction on F than not being a sunflower, it is plausible that (3.8) could be improved for non-trivial F Frankl and F¨uredi [7] did exactly this when they showed that (1.1) holds for all non-trivial 1-intersecting families We mentioned in the introduction that Stanton and Mullin [14] conjectured that (3.8) could

be improved to (1.2) if F is non-trivial and k-uniform; Theorem 3.1 verifies Stanton and Mullin’s conjecture for λ = 1 and Hall proved Stanton and Mullin’s conjecture when

λ = 2

then

2

 + 1

We adapt Hall’s proof of Theorem 3.2 to prove Theorem 1.3 (For the reader’s con-venience, we first reproduce Hall’s proof of Theorem 3.2.) In our proof of Theorem 1.3,

we will use the fact that if F ⊂ Xk
is a λ-intersecting family, then deg(x) does not lie in

then, for all x ∈ X,

McCarthy and Vanstone [12] adapted an argument of Connor [3], and improved this bound; they gave the following restriction on deg(x)

of size m

(i) If x ∈ X then,

(ii) Let {x, y} ⊂ X2
and define

The following determinant is non-negative:

det a11 a12

a21 a22



We now reproduce Hall’s proof of Theorem 3.2 Note that Hall had originally used (3.9) in his proof, but we will use (3.10) instead since it makes the argument cleaner

Hall’s Proof of Theorem 3.2 Suppose, for a contradiction, that there exists a non-trivial 2-intersecting F ⊂ Xk
of size m > k

2
+ 1 Write

2



Observe that the left hand side of (3.10) is quadratic in deg(x) with roots deg(x) = 0 and deg(x) = m − 1 + k/2 If there exists an x ∈ X with k ≤ deg(x) ≤ (m − 1 + k/2) − k, then (3.10) is true for deg(x) = k; together with (3.12), this implies that ǫ ≤ 0, which is impossible Hence, for all x ∈ X, either

2



We say a vertex x ∈ X with deg(x) ≤ k − 1 is light and is heavy if deg(x) ≥ m − ⌈k/2⌉

By (3.12), for any F ∈ F , we have

X

x∈F

Since the average degree of a vertex in F ∈ F is greater than k, every set F ∈ F contains

a heavy vertex As F is non-trivial, there are at least two heavy vertices x1, x2 Define

We have s ≤ k − 1 because λ = 2 and F is non-trivial Since u + v and t + v count

Consequently (3.12) implies,

k

2



2



≤ 2k

, we have a contradiction for k ≥ 5 For k = 4, Theorem 3.1 yields m ≤ 7, so

that does not satisfy (1.2) has at least λ heavy vertices, then Hall’s argument would yield a proof of Conjecture 1.5 Unfortunately, Hall’s averaging argument only shows that any non-trivial

proof of Theorem 1.3, we expend a lot of effort to eliminate the possibility that there are exactly two heavy vertices when λ = 3; the key difficulty is getting a good bound on the number of sets F ∈ F that contain both the heavy vertices

Proof of Theorem 1.3 We observe that Theorem 3.2 together with Theorem 1.2 yields Theorem 1.3 (i)

design First suppose k < 6 It is not difficult to see that if F is a non-trivial k-uniform 3-intersecting family of size m, where k ∈ {4, 5}, then m ≤ 5; for proofs of these results

in a more general setting see [8], [9], and [15] Hence, (1.2) holds when k < 6

family of size m for which (1.2) does not hold Write

Note that the left hand side of (3.10) is quadratic in deg(x) with roots deg(x) = 0 and deg(x) = m − 1 + k/3 If there exists an x ∈ X with k ≤ deg(x) ≤ (m − 1 + k/3) − k, then (3.10) is true for deg(x) = k; together with (3.15), this implies that ǫ ≤ 0, which is impossible Hence, for all x ∈ X, either

3



Following Hall [10], we say a vertex x ∈ X is light if deg(x) ≤ k − 1 and is heavy if deg(x) ≥ m − ⌈2k/3⌉

By (3.15), for any F ∈ F , we have

X

x∈F

Since the average degree of a vertex in F ∈ F is greater than k, every set F ∈ F contains

a heavy vertex As F is non-trivial, there are at least two heavy vertices We consider two cases, according to whether there are exactly two or greater than two heavy vertices

is non-trivial, there exists a set F1 ∈ F which contains x1 but not x2, and there exists a set F2 ∈ F which contains x2 but not x1 Let F1∩ F2 := {y1, y2, y3} Define

u := |{F ∈ F : x1 ∈ F, x/ 2 ∈ F }|,

and observe that m = s + t + u since every F ∈ F contains a heavy vertex By (3.16), we have t, u ≤ ⌈2k/3⌉ ≤ (2k + 2)/3

We now show how to obtain an upper bound on s in terms of k Observe that any

F ∈ F that contains {x1, x2} intersects F1\ {x1} in a subset of size two Consequently,

{x1,x2}⊂F

w∈F1\{x1}

then codeg ({x1, x2, w}) ≤ (k − 1)/2 Suppose F′, ˆF ∈ F are distinct sets in F that both contain {x1, x2, w} Since λ = 3, we see that the intersections of F′ and ˆF with F \{x1, x2}

Observe that if w ∈ F1\ {x1, y1, y2, y3} then F2 is a set in F that satisfies (3.19) We will

that satisfies (3.19) for w = yi

Subcase 1: For each yi ∈ F1∩ F2, there exists an F ∈ F that satisfies (3.19) for w = yi Applying (3.16) and (3.18) yields

k(k − 1)

(k − 1)2

 2k 3



2

+ 10k + 19

This implies that k2

− 14k − 7 + 12ǫ ≤ 0, which is a contradiction for k ≥ 15 since ǫ ≥ 1/3 For the remaining values of k, we refer the reader to the appendix

Subcase 2: There exists a yi ∈ F1∩ F2 for which no F ∈ F satisfies (3.19) for w = yi.

codeg ({x1, x2, yi}) ≤ k − 1 − (t + u) Suppose that every F ∈ F , not containing {x1, x2}, contains j of the elements {y1, y2, y3} where j ∈ {1, 2, 3} Applying (3.18) yields

k(k − 1)

2



 X

w∈F 1 \{x 1 }

codeg (x1, x2, w)



2



2

 + t + u

(k − 1 − j)(k − 1)

t + u 2

(k − 1 − j)(k − 1)

2k + 2 3

7 6



4k + 4 3

2

+ 5k + 8

since the penultimate expression in (3.21) is maximized when j = 1 This implies that (k − 1)(k − 8) + 12(ǫ − 1/3) ≤ 0, which is a contradiction for k ≥ 9

bound codeg (x1, x2, w) ≤ 7/2 for vertices w that are not one of the j special vertices in {y1, y2, y3} If we replace the weaker bound on codeg (x1, x2, w) by the tighter bound,

so we also get a contradiction in this case

Case 2: There are greater than two heavy vertices Let x1, x2, x3 be three heavy vertices Define

s := |{F ∈ F : {x1, x2, x3} ⊂ F }|, t := |{F ∈ F : x1 ∈ F, x2, x3 ∈ F }|,/

u := |{F ∈ F : x2 ∈ F, x1, x3 ∈ F }|,/ v := |{F ∈ F : x3 ∈ F, x1, x2 ∈ F }|,/

w := |{F ∈ F : x1, x2 ∈ F, x3 ∈ F }|,/ x := |{F ∈ F : x1, x3 ∈ F, x2 ∈ F }|,/

y := |{F ∈ F : x2, x3 ∈ F, x1 ∈ F }|,/ z := |{F ∈ F : x1, x2, x3 ∈ F }|./

3



by (3.16) As λ = 3 and F is non-trivial, we have s ≤ k − 2 Therefore (3.22) implies,

k(k − 1)

≤ s + (u + v + y + z) + (t + v + x + z) + (t + u + w + z)

3



This implies k2

− 10k + 3 + 3ǫ ≤ 0, so we have a contradiction for k ≥ 10 since ǫ ≥ 1/3 For the remaining values of k, we refer the reader to the appendix

F | ≥ m2
and equality holds if and only if F is a symmetric design

4 Appendix

Here, we collect some computations that are needed to verify Theorem 1.3 for small values

of k We regret that we could not prove Theorem 1.3 for all values of k The missing cases are k = 11, m = 40 in Case 1, Subcase 1 and k = 8, m = 20 in Case 2

Case Analysis for Case 1, Subcase 1:

If k = 14, then codeg (x1, x2, w) ≤ 6 for w ∈ F1\ {x1} so (3.18) yields s ≤ 39 Using this value for s in (3.20) yields ǫ < 0, which contradicts (3.15)

If k = 13, then the penultimate inequality in (3.20) yields that m = 54, s = 36, and

t = u = 9 Using these values in (3.11) yields a contradiction

If k = 12, then codeg (x1, x2, w) ≤ 5 for w ∈ F1\ {x1} so (3.18) yields s ≤ 27 Using this value for s in (3.20) yields ǫ < 0, which contradicts (3.15)

(2.3), then the dual linear program becomes

2



2



2



w + ix ≤ 1, i ∈ {1, , m} \ {s}

s 2



w + sx + y ≤ 1

F | If m = 38, then (3.20)

−1

3
≤ q

2

−1

3
if

p > q ≥ 2 We have that 222

−1

3
+ 22 2

3
+ 631